Mathematics and the Real World (9 page)

This type of proof is known as
reduction ad absurdum
. An argument that made use of contradiction was not only exceptionally innovative in its time, but it also runs counter to the natural way in which the human brain works. How could a claim that starts with the words “Assume that X is not…” be developed? The reader is invited to try to remember when and where, apart from in mathematics lessons, he or she intuitively used a thought process that assumed that something did not exist. Intuitive thought is founded on association, on links between current observation and the recognition of previous situations. A nonexistent event does not naturally arise as an association. After so many years of mathematical developments it is hard to assess how revolutionary this approach was. The real reason for the Pythagoreans hiding the discovery of irrational numbers may have been that they were not completely sure of the validity of proof via reduction ad absurdum. The uncertainty concerning such proofs reappeared in modern times, and we will discuss it in the chapter on the foundations of mathematics.

The Pythagoreans were familiar with prime numbers, that is, the natural numbers that can be divided without a remainder by only themselves and
1, and they studied them extensively. Apart from anything else, the Greeks proved that the number of prime numbers is infinite. The proof is simple.

1. First, note that every number can be expressed as the product of prime factors of the number.
   
2. Multiply
   n
   prime numbers and add 1 to the product. We get a number that we denote by
   M
   .
   
3. If
   M
   is a prime number, we have found one not among the
   n
   prime numbers that we had started with.
   
4. If
   M
   is not prime, consider a prime factor of
   M
   .
   
5. The prime factor can be divided into
   M
   without a remainder, hence the prime factor of
   M
   is different from each of the
   n
   prime numbers we started with as these, when divided into
   M
   , give a remainder 1.
   
6. Thus, in the second possibility (i.e., step d), we also found another prime number in addition to the
   n
   numbers that we multiplied.
   
7. We have thus shown that there are more than
   n
   prime numbers. But
   n
   is an arbitrary number, therefore the number of prime numbers is not finite and the proof is complete.
   

But why should anyone be interested in the question of whether the number of prime numbers is infinite? Where in evolution would the question whether there is an infinite number of any particular object be a meaningful question? Interest in the mathematical properties of prime numbers, including apparently useless properties, started with the Greeks, continued throughout generations of mathematicians, and is still today an important part of mathematical research. In current times uses of prime numbers have been discovered apart from the abstract mathematical interest, including commercial uses such as encoding, which we will discuss further on. For thousands of years the interest was purely mathematical. For the Greeks, however, it seems that the involvement in numbers was not simply motivated by curiosity but by the belief that thus they would better understand the world around them.

The next leap in the amazing change wrought by the Greeks in the development of mathematics is attributed to the Academy of Athens and its disciples, and in particular to its founder, Plato, his friend Eudoxus, and Plato's pupil Aristotle. The conceptual contribution of this group may be summarized as the formulation of the approach that bases mathematics on axioms and on logic as the essential tool in the system of deductive proof. As we will try to establish here and later in this book, these two contributions conflict with the natural intuition of human thought.

Plato (427–347 BCE) came from an aristocratic and influential family. He was a pupil of Socrates, who is considered the father of general and political Western philosophy. In his youth, Plato entertained political ambitions, but he abandoned them, perhaps because he saw what happened to Socrates, who was sentenced to death for his opposition to and criticism of the rulers of Athens. Plato traveled widely in the ancient world, visiting Egypt and the Greek colonies in Sicily, where he became acquainted with Egyptian mathematics and the Pythagoreans. On returning to Athens, he founded the first academy in the Western world. The academy had a decisive influence on contemporary science and philosophy. Plato was essentially a philosopher, and his interest in mathematics stemmed from his belief that the truth about the nature of science can be revealed only via mathematics. On the entrance of the academy he inscribed “Let none but geometers enter here.” Plato went further and, in accordance with the philosophy he developed in other fields, claimed that mathematics, or mathematical results, have an independent existence in the world of ideas that are not necessarily related to the earthly reality that we experience in daily life. Specifically, we do not invent mathematical results; we discover them. The right way to do this is to formulate the assumptions, which we will call axioms, and to use them to draw mathematical truths from them using deductive logic. To this end the axioms should be simple and self-explanatory. The smaller the number of axioms, the better. In modern mathematics it is generally accepted (admittedly, not by all researchers) that the researcher can choose his axioms freely. It is doubtful whether Plato would agree. He believed that the axioms are the link between man and mathematical truth, and they must therefore be “correct.”

Formulating axioms and examining the situation under prescribed assumptions is today an accepted method of analysis not only in mathematics but also in many other fields. We should be aware, however, that this method is contrary to natural human thought. It is difficult to understand how evolution gives an advantage to someone who says, “I cannot see a tiger nearby and so I assume there is no tiger in the area.” What benefit will accrue for someone who ignores certain attributes just because he has not assumed them? Even mathematicians reared on the system
based on axioms cannot limit their intuition to axioms. First, they solve the problem confronting them intuitively, or guess how to solve it, and only then do they check whether their solution was based solely on the axioms or also on additional assumptions or on properties not consistent with the axioms. In the latter case they must look for another solution.

In Plato's time great emphasis was placed on abstract mathematical problems such as squaring the circle, dividing an angle into three equal angles, or doubling a cube (i.e., calculating the edge of a cube whose volume is double that of a given cube), and all these using only a ruler and a compass. In other words, they were attempting to draw a square with the same area as a given circle, using only a ruler and a compass, and likewise with the other problems.

These problems were known before Plato. The problem of squaring the circle is attributed to the philosopher Anaxagoras, who thought about it while in prison, charged with impiety. The problems received even greater attention in the time of Plato, and this was at the same time as efforts were made to base mathematical proofs on as few assumptions as possible. The answer to the problems was that it is impossible to perform those tasks with just a ruler and a compass. The complete proof was not obtained until the nineteenth century. These and similar problems motivated research from the time of the Greeks until today.

The question arises, what made the Greeks interested in these questions? One story ascribes the problem of the doubling of the volume of a cube to a Greek ruler who was envious of his counterpart in a neighboring city and asked the builders of a mausoleum in that city to build one for him with double the volume. The story is not very convincing. No reasonable ruler would restrict his builders to the use of only a ruler and a compass. The origin of the idea not to use all the means available does not
lie in the evolutionary struggle. Imagine ancient man fleeing from a tiger and thinking, “I wonder if I can run away on one leg only.” Such individuals would not survive. Another version of the source of these problems is found in the writings of Plutarch. According to this story, the inhabitants of the city of Delos asked the local oracle for advice as to how to stop the constant disputes among themselves. The oracle's answer (not surprising, it may be said) was that they should double the volume of Apollo's altar in the city. The inhabitants asked Plato how to do this, and he, arguing that the oracle doubtless had proper mathematical intentions, interpreted the oracle's instructions to mean they should build with the use of just a ruler and a compass (also not surprising, as Plato wanted to promote his method). The point is that whichever version one prefers, questions that apparently do not relate to practical problems have occupied mathematicians since then.

Eudoxus (408–355 BCE) was born in the Greek city of Knidus on Cyprus and studied under the Pythagorean mathematician Archytas. Eudoxus traveled to Egypt to study and in 368 BCE joined Plato's academy in Athens. Eudoxus made many contributions to astronomy and mathematics. Here we will concentrate on only some of his significant contributions to the philosophy and practice of mathematics. Two of his innovations derived from his study of irrational numbers. Already in his time, many geometric dimensions were known that could not be expressed as a ratio of two integers (Plato had shown that the square roots of the prime numbers up to 17 were irrational numbers). Today we refer to both rational and irrational numbers as numbers, but in those days it was not clear in what sense irrational numbers were numbers. Eudoxus developed a mathematical theory that drew a distinction between numbers used to count individual elements (today referred to as natural numbers) and their ratios on the one hand, and numbers that measure geometrical lengths on the other. According to Eudoxus, mathematical operations in the two systems have different meanings. The interpretation of operations like addition, multiplication, and so on, on geometrical dimensions, is geometric. For example, multiplying √2 by √3 expresses the area of a rectangle with sides of lengths √2 and √3.
With regard to natural numbers, Eudoxus defined the ratio of two numbers, say
n
divided by
m
as the number of times
m
goes into
n
, and the product of two numbers, again say
n
and
m
, as the result of counting
n
elements
m
times. The result of this distinction was the separation of geometry from algebra or arithmetic, a separation that was bridged only in the seventeenth century by René Descartes. It is interesting to note that even today we use the geometric concept of “square” to express a number multiplied by itself. The need to distinguish between two types of numbers resulted in Eudoxus's using concepts that still today are considered the foundations of mathematics, that is,
definitions
and
axioms
. He defined rational numbers, a point, a line, length, and so on, and formulated several axioms precisely. This was apparently one of the first attempts to present precise definitions and axioms.

The need to give exact definitions is not a natural one. In discussions between humans, it is sufficient to reach a situation in which the participants know what is being discussed, and it is not necessary to waste time on exact definitions of the topic under discussion. To invest time and effort in defining what is a point, a length, or a plane, when everyone knows what these terms mean, seems superfluous. The process of making a definition, apparently, seemed unnecessary also during the thousands of years of mathematical development prior to the Greeks, but it did not seem so to the academicians of Athens, and mathematics inherited the practice from them and has preserved it until today.

Another contribution made by Eudoxus, essentially technical, is known as the method of exhaustion. This was an extension of a development by the Pythagoreans, who, after discovering that irrational numbers cannot be expressed by means of natural numbers, showed that they can be approximated by ratios of natural numbers. Eudoxus developed a system for calculating areas enclosed by general curves, such as a circle, by removing the areas within them, such as rectangles or other shapes whose areas are simple to calculate, until the total area to be calculated is “exhausted.” Thus the area can be calculated by a close approximation. Eudoxus was very close to the concept of a limit, but he did not actually reach it. That was developed many years later by Archimedes, and is still used today as
the basis of differential and integral calculus. Over and above the intrinsic technical contribution of the exhaustion method, the very presentation and formulation of the general method was itself an important contribution.